Abstract Swelling is an important property of charged gels and membranes. The size of pores and, therefore, membrane properties such as conductivity, diffusion and hydraulic permeability, permselectivity depend on water content and degree of swelling. In this paper we propose a simple model for equilibrium swelling of an ion-exchange membrane, allowing calculation of water content and membrane thickness as functions of the concentration and pH of the bathing solution. The model parameters include the equivalent volume of dry polyelectrolyte gel, the volume fraction of macropores and others. Three types of ion-exchange functional groups, namely, the secondary, tertiary and quaternary amino groups in an anion-exchange membrane are taken into account. The osmotic pressure exerted by micro- and mesopores, appearing in the gel when swelling, is expressed using the Gregor equation, which employs the mole fractions of free and bound water. The equilibria between protonated and deprotonated amino groups are assumed as well as the Donnan and ion-exchange equilibria between the membrane and bathing solution. The results of calculations are compared with experimental data on the membrane thickness and effective exchange capacity obtained for two heterogeneous anion-exchange membranes MA-40 and MA-41 (Shchekinoazot) differed by the composition of functional groups. The procedure of determining membrane structural and thermodynamic parameters is described. A good quantitative agreement between the theory and experiment is found for both membranes using the same set of ion hydration numbers and chemical equilibrium constants for secondary and tertiary amino groups. In particular, it is shown that with increasing pH of the bathing solution, the membrane thickness (and, hence, water content) pass through a local maximum and a local minimum. The simplicity of the model, which does not reduce adequacy, would enable it to be included later in other models describing the transport of ions and water to account for the swelling and change in the membrane structure with a change in pH and concentration of the bathing solution.

中文翻译：

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阴离子交换膜对沐浴液浓度和 pH 值变化的响应的简单模型

摘要 溶胀是带电凝胶和膜的重要特性。孔的大小以及因此的膜特性（例如电导率、扩散和水力渗透率、渗透选择性）取决于含水量和溶胀程度。在本文中，我们提出了一个简单的离子交换膜平衡溶胀模型，允许计算水含量和膜厚度作为沐浴液浓度和 pH 值的函数。模型参数包括干聚电解质凝胶的等效体积、大孔的体积分数等。考虑了三种类型的离子交换官能团，即阴离子交换膜中的仲氨基、叔氨基和季氨基。微孔和中孔施加的渗透压，在溶胀时出现在凝胶中，使用 Gregor 方程表示，该方程使用自由水和结合水的摩尔分数。假设质子化和去质子化氨基之间的平衡以及膜和沐浴液之间的唐南和离子交换平衡。将计算结果与两种异质阴离子交换膜 MA-40 和 MA-41 (Shchekinoazot) 获得的膜厚度和有效交换容量的实验数据进行比较，这两种膜因官能团的组成而异。描述了确定膜结构和热力学参数的过程。对于使用相同组离子水合数和仲氨基和叔氨基化学平衡常数的两种膜，发现理论和实验之间的定量一致。特别是，结果表明，随着沐浴液 pH 值的增加，膜厚度（以及因此的含水量）通过局部最大值和局部最小值。该模型的简单性不会降低充分性，因此可以稍后将其包含在描述离子和水传输的其他模型中，以解释膜结构随 pH 值和浓度变化而发生的膨胀和变化解决方案。